I'm assuming you mean strength and not thickness. A higher strength means that the reed is more resistant, so more air must be used to make a sound. Generally beginners start on a low strength and move up as you build your embouchure, but a higher strength doesn't mean you're a better player than someone on a lower strength. As for tone, a higher strength usually offers a more controlled tone but with less projection. People usually play stronger reeds for classical and softer reeds for jazz/rock/pop.
Harder reed = less flexible reed. Softer reed = more flexible reed. Literally, the tip will be easier to bend. Generally mouthpieces with a more closed tip and longer facing require a stronger reed, otherwise it is easy to "pinch" the reed closed. As the tip openings get bigger and with a shorter facing length, softer reeds are needed. This is a general explanation - I tend to use a slightly harder reed on a open tip but the facing length is long. Hope you're not confused yet. Softer reeds may give a fuller tone but at the expense of control and upper register ease.
In general, soft reeds tend to vibrate easier, are easier to blow and more flexible, but you need some strength to get a core to the sound and control in the upper register. Hard reeds tend to produce a dark (e.g., less vibrant, less resonant, less responsive) sound. In my opinion, the trick is to find a reed no harder than needed to do everything you want, with the tone quality you want. Every person and every clarinet/mouthpiece/reed setup is different. As a previous poster explained, it's the density of the reed that seems to mostly determine the strength.
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